Abstract
Three characterizations of quasi-median graphs are proved, for instance, they are partial Hamming graphs without convex house and convex Q 3 − such that certain relations on their edge sets coincide. Expansion procedures, weakly 2-convexity, and several relations on the edge set of a graph are essential for these results. Quasi-semimedian graphs are characterized which yields an additional characterization of quasi-median graphs. Two equalities for quasi-median graphs are proved. One of them asserts that if α i , i≥0, denotes the number of induced Hamming subgraphs of a quasi-median graph, then ∑ i≥0 (−1) i α i=1 . Finally, an Euler-type formula is derived for graphs that can be obtained by a sequence of connected expansions from K 1.
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